

Setting \(\kappa =0\) , the scheme (REF ) reduces exactly to the original ESAV scheme (without stabilization) presented in [1]}.
The Crank–Nicolson type ESAV scheme (ESAV2) is given by
\(& \delta _t u^{n+1} = \varepsilon ^2\Delta _h u^{n+\frac{1}{2}} - \kappa u^{n+\frac{1}{2}}+ \frac{\widehat{r}^{n+\frac{1}{2}}}{\exp \lbrace E_{2h}(\widehat{u}^{n+\frac{1}{2}})\rbrace }(f(\widehat{u}^{n+\frac{1}{2}})+\kappa \widehat{u}^{n+\frac{1}{2}}), \\& \frac{\ln r^{n+1} - \ln r^n}{{\tau }} = - \frac{\widehat{r}^{n+\frac{1}{2}}}{\exp \lbrace E_{2h}(\widehat{u}^{n+\frac{1}{2}})\rbrace }\langle f(\widehat{u}^{n+\frac{1}{2}})+\kappa \widehat{u}^{n+\frac{1}{2}},\delta _t u^{n+1}\rangle ,\) 
